Thermodynamics - Specific heats at constant pressure, volume, Specific heat ratio, Gas constants

Specific heat

The ratio of the amount of heat required to raise the temperature of a unit mass of a substance by one unit of temperature to the amount of heat required to raise the temperature of a similar mass of a reference material, usually water, by the same amount.


Specific heat ratio

The specific heat ratio of a gas is the ratio of the specific heat at constant pressure, Cp, to the specific heat at constant volume, Cv. It is sometimes referred to as the adiabatic index or the heat capacity ratio or the isentropic expansion factor or the adiabatic exponent or the isentropic exponent.

For an ideal gas, the heat capacity is constant with temperature.


k = Enthalpy/Internal energy

k = H/U


Enthalpy

H = Cp * T


Internal energy

U = Cv * T


Specific heat ratio (k)

k = H/U

k = Cp * T/CV * T

k = Cp/Cv


Specific heat and Gas constant R

Cp = Cv + R

R = Cp - Cv



---Derivation---

Heating a gas at constant pressure increases the internal energy of the gas and Work is done, whereas supplying the same amount of heat at constant volume only increases the internal energy, no work is done.



constant pressure process:

du = dq - w

dq = du + w

w = pdV

dq = du + pdV


from ideal gas relations

pV = mRT


but at constant pressure

pdV = mRdT


thus

dq = du + mRdT ---> equation1


and

dq = m * cp * dT


equation1 becomes

m * cp * dT = du + mRdT ---> equation2





constant volume process:

du = dq - w


at constant volume, w = 0

w = pdV

w = p(v2 - v1)

but v2 = v1

w = p(0)

w = 0


du = dq + 0

du = dq

dq = m * Cv * dT

du = m * Cv * dT ---> equation3


equation3 in equation2

m * cp * dT = du + mRdT

m * cp * dT = (m * Cv * dT) + mRdT


factoring

(m * dT) (Cp) = (m * dT) (Cv + R)


(m * dT) cancels and thus leaving

Cp = Cv + R